Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Two-Sided Matching Models

  • Marilda Sotomayor
  • Ömer Özak
Reference work entry

Article Outline


Definition of the Subject


Discrete Two-Sided Matching Models

Continuous Two-Sided Matching Model with Additively Separable Utility Functions

Hybrid One-to-One Matching Model


Future Directions



Nash Equilibrium Competitive Equilibrium Match Model Marriage Market Stable Matchings 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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  1. 1.
    Abdulkadiroglu A, Sönmez T (2003)School Choice: A Mechanism Design Approach.Am Econ Rev 93(3):729–747Google Scholar
  2. 2.
    Abdulkadiroglu A, Pathak P, Roth A, Sönmez T (2006)Changing the Boston School Choice Mechanism: Strategy‐proofness as Equal Access, working paper.Boston College and Harvard University, BostonGoogle Scholar
  3. 3.
    Abeledo H, Isaak G (1991)A characterization of graphs which assure the existence of stable matchings.Math Soc Sci 22(1):93–96MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Adachi H (2000)On a characterization of stable matchings.Econ Lett 68(1):43–49MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Alcalde J (1996)Implementation of Stable Solutions to Marriage Problems.J Econ Theory 69(1):240–254MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Alcalde J, Pérez‐Castrillo D, Romero‐Medina A (1998)Hiring Procedures to Implement Stable Allocations.J Econ Theory 82(2):469–480Google Scholar
  7. 7.
    Alcalde J, Romero‐Medina A (2000)Simple Mechanisms to Implement the Core of College Admissions Problems.Games Econ Behav 31(2):294–302Google Scholar
  8. 8.
    Balinski M, Sönmez T (1999)A Tale of Two Mechanisms: Student Placement.J Econ Theory 84(1):73–94Google Scholar
  9. 9.
    Bardella F, Sotomayor M (2006)Redesign and analysis of an admission market to the graduate centers of economics in Brazil:a natural experiment in market organization, working paper.Universidade de São Paulo, São PauloGoogle Scholar
  10. 10.
    Bikhchandani S, Mamer J (1997)Competitive Equilibrium in an Exchange Economy with Indivisibilities.J Econ Theory 74(2):385–413MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Birkhoff G (1973)Lattice theory, vol v. 25. Colloquium publications.American Mathematical Society, ProvidenceGoogle Scholar
  12. 12.
    Blair C (1988)The Lattice Structure of the Set of Stable Matchings with Multiple Partners.Math Oper Res 13(4):619–628MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chung K (2000)On the Existence of Stable Roommate Matchings.Games Econ Behav 33(2):206–230zbMATHCrossRefGoogle Scholar
  14. 14.
    Crawford V (1991)Comparative statics in matching markets.J Econ Theory 54(2):389–400zbMATHCrossRefGoogle Scholar
  15. 15.
    Crawford V (2008)The Flexible‐Salary Match: A Proposal to Increase the Salary Flexibility of the National Resident Matching Program.J Econ Behav Organ 66(2):149–160MathSciNetCrossRefGoogle Scholar
  16. 16.
    Crawford V, Knoer E (1981)Job Matching with Heterogeneous Firms and Workers.Econometrica 49(2):437–450zbMATHCrossRefGoogle Scholar
  17. 17.
    Demange G (1982)Strategyproofness in the assignment market game, working paper.Ecole Polytechnique, Laboratoire D'Econometrie, ParisGoogle Scholar
  18. 18.
    Demange G, Gale D (1985)The Strategy Structure of Two-Sided Matching Markets.Econometrica 53(4):873–888MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Demange G, Gale D, Sotomayor M (1986)Multi-Item Auctions.J Political Econ 94(4):863–872CrossRefGoogle Scholar
  20. 20.
    Demange G, Gale D, Sotomayor M (1987) A further note on the stable matchingproblem.Discret Appl Math 16(3):217–222MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Dubins L, Freedman D (1981)Machiavelli and the Gale‐Shapley Algorithm.Am Math Mon 88(7):485–494MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dutta B, Massó J (1997)Stability of Matchings When Individuals Have Preferences over Colleagues.J Econ Theory 75(2):464–475Google Scholar
  23. 23.
    Echenique F, Oviedo J (2004)Core many-to-one matchings by fixed-point methods.J Econ Theory 115(2):358–376MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Echenique F, Oviedo J (2006)A theory of stability in many-to-many matching markets.Theor Econ 1(2):233–273Google Scholar
  25. 25.
    Echenique F, Yenmez M (2007)A solution to matching with preferences over colleagues.Games Econ Behav 59(1):46–71MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Eeckhout J (2000)On the uniqueness of stable marriage matchings.Econ Lett 69(1):1–8MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ehlers L, Massó J (2004)Incomplete Information and Small Cores in Matching Markets, working paper. CREA, BarcelonaGoogle Scholar
  28. 28.
    Ergin H, Sönmez T (2006)Games of school choice under the Boston mechanism.J Public Econ 90(1–2):215–237Google Scholar
  29. 29.
    Eriksson K, Karlander J (2000)Stable matching in a common generalization of the marriage and assignment models.Discret Math 217(1):135–156MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Gale D, Shapley L (1962)College Admissions and the Stability of Marriage.Am Math Mon 69(1):9–15MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Gale D, Sotomayor M (1985)Ms. Machiavelli and the Stable Matching Problem.Am Math Mon 92(4):261–268MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Gale D, Sotomayor M (1983,1985)Some remarks on the stable matching problem.Discret Appl Math 11:223–232Google Scholar
  33. 33.
    Gül F, Stacchetti E (1999)Walrasian Equilibrium with Gross Substitutes.J Econ Theory 87(1):95–124Google Scholar
  34. 34.
    Gül F, Stacchetti E (2000)The English Auction with Differentiated Commodities.J Econ Theory 92(1):66–95Google Scholar
  35. 35.
    Gusfield D (1988)The Structure of the Stable Roommate Problem: Efficient Representation and Enumeration of All Stable Assignments.SIAM J Comput 17:742–769MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hatfield J, Milgrom P (2005)Matching with Contracts.Am Econ Rev 95(4):913–935CrossRefGoogle Scholar
  37. 37.
    Irving R (1985)An efficient algorithm for the stable roommates problem.J Algorithms 6:577–595MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kamecke U (1989)Non‐cooperative matching games.Int J Game Theory 18(4):423–431MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kaneko M (1982)The central assignment game and the assignment markets.J Math Econ 10(2–3):205–232zbMATHCrossRefGoogle Scholar
  40. 40.
    Kara T, Sönmez T (1996)Nash Implementation of Matching Rules.J Econ Theory 68(2):425–439Google Scholar
  41. 41.
    Kara T, Sönmez T (1997)Implementation of college admission rules.J Econ Theory 9(2):197–218Google Scholar
  42. 42.
    Kelso Jr A, Crawford V (1982)Job Matching, Coalition Formation, and Gross Substitutes.Econometrica 50(6):1483–1504zbMATHCrossRefGoogle Scholar
  43. 43.
    Kesten O (2004)Student placement to public schools in the US: Two new solutions, working paper.University of Rochester, RochesterGoogle Scholar
  44. 44.
    Kesten O (2006)On two competing mechanisms for priority‐based allocation problems.J Econ Theory 127(1):155–171MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Knuth D (1976)Marriage Stables.Les Presses de l'Université de Montréal, MontréalGoogle Scholar
  46. 46.
    KonishiH, Ünver MU (2006)Credible group stability in many-to-many matching problems.J Econ Theory 127(1):57–80Google Scholar
  47. 47.
    Kraft C, Pratt J, Seidenberg A (1959)Intuitive Probability on Finite Sets.Ann Math Stat 30(2):408–419MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Leonard H (1983)Elicitation of Honest Preferences for the Assignment of Individuals to Positions.J Political Econ 91(3):461–479CrossRefGoogle Scholar
  49. 49.
    Ma J (2002)Stable matchings and the small core in Nash equilibrium in the college admissions problem.Rev Econ Des 7(2):117–134zbMATHCrossRefGoogle Scholar
  50. 50.
    Martínez R, Massó J, Neme A, Oviedo J (2001)On the lattice structure of the set of stable matchings for a many-to-one model.Optimization 50(5):439–457Google Scholar
  51. 51.
    Martínez R, Massó J, Neme A, Oviedo J (2004)An algorithm to compute the full set of many-to-many stable matchings.Math Soc Sci 47(2):187–210Google Scholar
  52. 52.
    McVitie D, Wilson L (1970)Stable marriage assignment for unequal sets.BIT Numer Math 10(3):295–309zbMATHCrossRefGoogle Scholar
  53. 53.
    Mo J (1988)Entry and structures of interest groups in assignment games.J Econ Theory 46(1):66–96zbMATHCrossRefGoogle Scholar
  54. 54.
    Ostrovsky M (2008) Stability in Supply Chain Networks. Am Econ Rev98(3):897–923MathSciNetCrossRefGoogle Scholar
  55. 55.
    Pathak P, Sönmez T (2006) Leveling the Playing Field: Sincere and Strategic Playersin the Boston Mechanism, working paper. Boston College and Havard University, BostonGoogle Scholar
  56. 56.
    Pérez‐Castrillo D, Sotomayor M (2002)A Simple Selling and Buying Procedure.J Econ Theory 103(2):461–474Google Scholar
  57. 57.
    Pycia M (2007) Many-to-One Matching with Complementarities and Peer Effects, workingpaper. Penn State Working Paper, PennsylvaniaGoogle Scholar
  58. 58.
    Rochford S (1984)Symmetrically Pairwise‐Bargained Allocations in an Assignment Market.J Econ Theory 34(2):262–281MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Romero‐Medina A (1998)Implementation of stable solutions in a restricted matching market.Rev Econ Des 3(2):137–147Google Scholar
  60. 60.
    Roth A (1982)The Economics of Matching: Stability and Incentives.Math Oper Res 7(4):617–628MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Roth A (1984)Misrepresentation and Stability in the Marriage Problem.J Econ Theory 34(2):383–387zbMATHCrossRefGoogle Scholar
  62. 62.
    Roth A (1984)Stability and Polarization of Interests in Job Matching.Econometrica 52(1):47–58zbMATHCrossRefGoogle Scholar
  63. 63.
    Roth A (1984)The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory.J Political Econ 92(6):991–1016CrossRefGoogle Scholar
  64. 64.
    Roth A (1985)Conflict and Coincidence of Interest in Job Matching: Some New Results and Open Questions.Math Oper Res 10(3):379–389MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Roth A (1985)The College Admissions Problem is not Equivalent to the Marriage Problem.J Econ Theory 36(2):277–288zbMATHCrossRefGoogle Scholar
  66. 66.
    Roth A (1986)On the Allocation of Residents to Rural Hospitals: A General Property of Two-Sided Matching Markets.Econometrica 54(2):425–427MathSciNetCrossRefGoogle Scholar
  67. 67.
    Roth A, Sotomayor M (1988)Interior points in the core of two-sided matching markets.J Econ Theory 45(1):85–101MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Roth A, Sotomayor M (1989)The College Admissions Problem Revisited.Econometrica 57(3):559–570MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Roth A, Sotomayor M (1990)Two-Sided Matching: A Study in Game‐Theoretic Modeling and Analysis, vol 18. In: Econometric SocietyMonographs. Cambridge University Press, New YorkGoogle Scholar
  70. 70.
    Roth A, Sotomayor M (1996)Stable outcomes in discrete and continuous models of two-sided matching: A unified treatment.Braz Rev Econom 16:1–4Google Scholar
  71. 71.
    Shapley L (1962)Complements and substitutes in the Optimal Assignment Problem.Navals Res Logist Q 9:45–48CrossRefGoogle Scholar
  72. 72.
    Shapley L, Shubik M (1972)The Assignment Game I: The core.Int J Game Theory 1(1):111–130MathSciNetCrossRefGoogle Scholar
  73. 73.
    Sönmez T (1999)Strategy‐Proofness and Essentially Single‐Valued Cores.Econometrica 67(3):677–689Google Scholar
  74. 74.
    Sotomayor M (1986)On incentives in a two-sided matching market, working paper.Depart of Mathematics, PUC/RJ, Rio de JaneiroGoogle Scholar
  75. 75.
    Sotomayor M (1992)The multiple partners game in Equilibrium and dynamics.In: Majumdar M (ed) Essays in honour of David Gale.MacMillan Press Ltd, New York, pp 322–336Google Scholar
  76. 76.
    Sotomayor M (1996)A Non‐constructive Elementary Proof of the Existence of Stable Marriages.Games Econ Behav 13(1):135–137MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Sotomayor M (1996)Admission mechanisms of students to colleges. A game‐theoretic modeling and analysis.Braz Rev Econom 16(1):25–63MathSciNetGoogle Scholar
  78. 78.
    Sotomayor M (1998) The strategy structure of the college admissions stablemechanisms.In: Annals of Jornadas Latino Americanas de Teoria Econômica, San Luis, Argentina, 2000; First World Congress of the Game Theory Society, Bilbao,2000; 4th Spanish Meeting, Valencia, 2000; International Symposium of Mathematical Programming, Atlanta, 2000; World Congress of the Econometric Society,Seattle, 2000,
  79. 79.
    Sotomayor M (1999)The lattice structure of the set of stable outcomes of the multiple partners assignment game.Int J Game Theory 28(4):567–583MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Sotomayor M (1999)Three remarks on the many-to-many stable matching problem.Math Soc Sci 38(1):55–70MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Sotomayor M (2000)Existence of stable outcomes and the lattice property for a unified matching market.Math Soc Sci 39(2):119–132MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Sotomayor M (2000) Reaching the core through college admissions stable mechanisms.In: Annals of abstracts of the following congresses: International Conference on Game Theory, 2001, Stony Brook; Annals of Brazilian Meeting of Econometrics,Salvador, Brazil, 2001; Latin American Meeting of the Econometric Society, Buenos Aires, Argentina, 2001,
  83. 83.
    Sotomayor M (2002)A simultaneous descending bid auction for multiple items and unitary demand.Revista Brasileira Economia 56:497–510Google Scholar
  84. 84.
    Sotomayor M (2003)A labor market with heterogeneous firms and workers.Int J Game Theory 31(2):269–283MathSciNetCrossRefGoogle Scholar
  85. 85.
    Sotomayor M (2003)Reaching the core of the marriage market through a non‐revelation matching mechanism.Int J Game Theory 32(2):241–251MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Sotomayor M (2003)Some further remark on the core structure of the assignment game.Math Soc Sci 46:261–265MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Sotomayor M (2004) Buying and selling strategies in the assignment game, workingpaper. Universidade São Paulo, São PauloGoogle Scholar
  88. 88.
    Sotomayor M (2004)Implementation in the many-to-many matching market.Games Econ Behav 46(1):199–212MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Sotomayor M (2005) The roommate problem revisited, working paper.Universidade SãoPaulo, São Paulo.
  90. 90.
    Sotomayor M (2006) Adjusting prices in the many-to-many assignment game to yield thesmallest competitive equilibrium price vector, working paper. Universidade de São Paulo, SãoPaulo.
  91. 91.
    Sotomayor M (2007)Connecting the cooperative and competitive structures of the multiple partners assignment game.J Econ Theory 134(1):155–174MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Sotomayor M (2007)Core structure and comparative statics in a hybrid matching market.Games Econ Behav 60(2):357–380MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Sotomayor M (2008) The Pareto-Stability concept is a natural solution concept forthe Discrete Matching Markets with indifferences, working paper. Universidade São Paulo, SãoPaulo.
  94. 94.
    Sotomayor M (2007) The stability of the equilibrium outcomes in the admission gamesinduced by stable matching rules. Inter Jour Game Theory 36(3–4):621–640MathSciNetGoogle Scholar
  95. 95.
    Tan J (1991)A Necessary and Sufficient Condition for the Existence of a Complete Stable Matching.J Algorithms 12(1):154–178MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Tarski A (1955)A lattice‐theoretical fixpoint theorem and its applications.Pac J Math 5(2):285–309MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Thompson G (1980)Computing the core of a market game.In: Fiacco AA, Kortane K (eds) Extremal Methods and Systems Analysis, vol 174.Springer, New York, pp 312–324CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Marilda Sotomayor
    • 1
    • 2
  • Ömer Özak
    • 2
  1. 1.Department of EconomicsUniversity of São Paulo/SPSão PauloBrazil
  2. 2.Department of EconomicsBrown UniversityProvidenceUSA